MinT - Best Known (67−56, 67, s)-Nets in Base 5

Best Known (67−56, 67, s)-Nets in Base 5

(67−56, 67, 32)-Net over F₅ — Constructive and digital

Digital (11, 67, 32)-net over F₅, using

net from sequence [i] based on digital (11, 31)-sequence over F₅, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 11 and N(F) ≥ 32, using
  - an explicitly constructive algebraic function field [i]

(67−56, 67, 63)-Net over F₅ — Upper bound on s (digital)

There is no digital (11, 67, 64)-net over F₅, because

6 times m-reduction [i] would yield digital (11, 61, 64)-net over F₅, but
- extracting embedded orthogonal array [i] would yield linear OA(5⁶¹, 64, F₅, 50) (dual of [64, 3, 51]-code), but
  - Griesmer bound [i]

(67−56, 67, 64)-Net in Base 5 — Upper bound on s

There is no (11, 67, 65)-net in base 5, because

8 times m-reduction [i] would yield (11, 59, 65)-net in base 5, but
- extracting embedded orthogonal array [i] would yield OA(5⁵⁹, 65, S₅, 48), but
  - the linear programming bound shows that MÂ â‰¥ 802 309607 639273 281165 515072 643756 866455 078125 / 3773 > 5⁵⁹ [i]